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### This function is a component of astrochron: An R Package for Astrochronology
### Copyright (C) 2021 Stephen R. Meyers
###
###########################################################################
### MTM function - (SRM: February 28, 2012; March 29, 2012;
### September 18, 2012; Oct 8, 2012; Oct. 12, 2012; Nov. 23, 2012
### May 20-21, 2013; May 23, 2013; May 27, 2013; June 5, 2013;
### June 13, 2013; August 5, 2013; August 12, 2013; Nov. 26, 2013;
### July 31, 2014; January 31, 2015; February 1-3, 2015;
### February 26, 2015; March 6, 2015; June 30, 2015; Sept. 10, 2015;
### December 14, 2015; May 20, 2016; August 22, 2016; Oct. 4, 2016;
### March 20, 2017; November 20, 2017; August 17, 2018;
### January 14, 2021)
###
### uses multitaper library and built in functions from R
###########################################################################
mtm <- function (dat,tbw=3,ntap=NULL,padfac=5,demean=T,detrend=F,siglevel=0.9,ar1=T,output=0,CLpwr=T,xmin=0,xmax=Nyq,pl=1,sigID=T,genplot=T,verbose=T)
{
if(verbose) cat("\n----- PERFORMING Multitaper Spectral Analysis -----\n")
dat <- data.frame(dat)
npts <- length(dat[,1])
dt <- dat[2,1]-dat[1,1]
# error checking
if(dt<0)
{
if (verbose) cat("\n * Sorting data into increasing height/depth/time, removing empty entries\n")
dat <- dat[order(dat[,1], na.last = NA, decreasing = F), ]
dt <- dat[2,1]-dat[1,1]
npts <- length(dat[,1])
}
dtest <- dat[2:npts,1]-dat[1:(npts-1),1]
epsm=1e-9
if( (max(dtest)-min(dtest)) > epsm )
{
cat("\n**** ERROR: sampling interval is not uniform.\n")
stop("**** TERMINATING NOW!")
}
if (verbose)
{
cat(" * Number of data points in stratigraphic series:",npts,"\n")
cat(" * Stratigraphic series length (space or time):",(npts-1)*dt,"\n")
cat(" * Sampling interval (space or time):",dt,"\n")
}
numtap=trunc((2*tbw)-1)
if(is.null(ntap))
{
ntap=numtap
if (verbose) cat(" * Will use default setting of",ntap,"DPSS tapers\n")
}
if(ntap > numtap)
{
ntap=numtap
if (verbose) cat("**** WARNING: The number of DPSS tapers specified is too large. ntap reset to default value of",ntap,"\n")
}
if(ntap < 2)
{
ntap=numtap
if (verbose) cat("**** WARNING: The number of DPSS tapers specified is too small. ntap reset to default value of",ntap,"\n")
}
###########################################################################
### MTM Power spectrum using 'multitaper' library
###########################################################################
### this version computes the adaptive multitaper spectrum
# remove mean and linear trend if requested
if (demean)
{
dave <- colMeans(dat[2])
dat[2] <- dat[2] - dave
if(verbose) cat(" * Mean value subtracted=",dave,"\n")
}
if (!demean && verbose) cat(" * Mean value NOT subtracted\n")
### use least-squares fit to remove linear trend
if (detrend)
{
lm.0 <- lm(dat[,2] ~ dat[,1])
dat[2] <- dat[2] - (lm.0$coeff[2]*dat[1] + lm.0$coeff[1])
if(verbose) cat(" * Linear trend subtracted. m=",lm.0$coeff[2],"b=",lm.0$coeff[1],"\n")
}
if (!detrend && verbose) cat(" * Linear trend NOT subtracted\n")
### calculate Nyquist freq
Nyq <- 1/(2*dt)
### calculate Rayleigh frequency
Ray <- 1/(dt*npts)
npad=as.integer(npts*padfac)
### add another zero if we don't have an even number of data points, so Nyquist exists.
if((npad*padfac)%%2 != 0) npad = npad + 1
# padded frequency grid
df = 1/(npad*dt)
if(verbose)
{
cat(" * Nyquist frequency:",Nyq,"\n")
cat(" * Rayleigh frequency:",Ray,"\n")
cat(" * MTM Power spectrum bandwidth resolution (halfwidth):",tbw/(npts*dt),"\n")
cat(" * Padded to",npad,"points\n")
}
# make dat a time series object, here with unit sampling interval
datTS<-as.ts(dat[,2])
spec <- spec.mtm(datTS,nw=tbw,k=ntap,Ftest=T,nFFT=npad,taper=c("dpss"),centre=c("none"),jackknife=F,returnZeroFreq=F,plot=F)
# assign correct frequencies to spec$freq (note: this variable is returned if output = 4)
spec$freq <- spec$freq/dt
# note: no zero frequency present, also remove Nyquist now
nfreq = length(spec$freq) - 1
freq <- spec$freq[1:nfreq]
# normalize power (divided by npts in spec.mtm)
pwrRaw <- spec$spec[1:nfreq]/npts
FtestRaw <- spec$mtm$Ftest[1:nfreq]
# AR(1) noise model spectrum
if(ar1)
{
### what is the estimated AR1 coefficient?
lag0 <- dat[1:(npts-1),2]
lag1 <- dat[2:npts,2]
rho_raw <- cor(lag0,lag1)
### Calculate Raw red noise spectrum
### "So" is the average power. This can be determined from the white noise variance
### as So = var/(1-rho^2), where rho is the lag-1 coeff
### We will determine average power directly from measured spectrum
So = mean(pwrRaw)
RawAR = So * (1-(rho_raw^2)) / ( 1 - (2*rho_raw*cos(pi*freq/Nyq)) + (rho_raw^2) )
dofAR = (2*ntap)
chiRawAR <- (pwrRaw/RawAR) * dofAR
chiCLRawAR <- pchisq(chiRawAR, df=dofAR)
### 90, 95 and 99% CL for power plot
if(CLpwr)
{
AR1_90 <- RawAR*qchisq(0.9, df=dofAR)/dofAR
AR1_95 <- RawAR*qchisq(0.95, df=dofAR)/dofAR
AR1_99 <- RawAR*qchisq(0.99, df=dofAR)/dofAR
}
}
### f-test CL
dof=2*ntap
prob <- pf(FtestRaw,2,dof-2)
###########################################################################
### Identify "significant" frequencies
###########################################################################
if(verbose)
{
cat("\n * Searching for significant spectral peaks that satisfy",siglevel*100,"% CL\n")
cat(" requirements outlined in Meyers (2012):\n")
}
### identify the harmonic f-test peaks
res <- peak(cbind(freq,prob),level=siglevel,genplot=F,verbose=F)
numpeak = length(res[,1])
freqloc = res[,1]
probmax = res[,3]
# Frequency and Harmonic_CL will be written over later if ar1=T
Frequency <- freq[freqloc]
Harmonic_CL <- prob[freqloc]
if(!ar1 && verbose)
{
cat(" * Number of significant F-test peaks identified =",numpeak,"\n")
cat("ID / Frequency / Period / Harmonic_CL \n")
for(i in 1:numpeak) cat(i," ", Frequency[i]," ",1/Frequency[i]," ",Harmonic_CL[i]*100,"\n")
}
# if ar1 selected, also evaluate red noise
if(ar1)
{
# FORTRAN wrapper
peakfilter <- function (numpeak,nfreq,tbwRay,siglevel,freqloc,probmax,freq,background,pwr,cl)
{
F_dat = .Fortran('peakfilter_r',
numpeak=as.integer(numpeak),nfreq=as.integer(nfreq),tbwRay=as.double(tbwRay),
siglevel=as.double(siglevel),freqloc=as.integer(freqloc),probmax=as.double(probmax),
freq=as.double(freq),background=as.double(background),pwr=as.double(pwr),
cl=as.double(cl),
loc=integer(as.integer(numpeak)), nout=integer(1)
)
# return the results
return(F_dat)
}
# identify maxima of peaks
tbwRay=tbw*Ray
res2 <- peakfilter(numpeak,nfreq,tbwRay,siglevel,freqloc,probmax,freq,RawAR,pwrRaw,chiCLRawAR)
numpeak2=res2$nout
loc=res2$loc[1:numpeak2]
Frequency <- freq[loc]
Harmonic_CL <- prob[loc]
Red_Noise_CL <- chiCLRawAR[loc]
if(verbose)
{
cat(" * Number of significant F-test peaks identified =",numpeak2,"\n")
cat("ID / Frequency / Period / Harmonic_CL / Rednoise_CL\n")
for(i in 1:numpeak2) cat(i," ", Frequency[i]," ",1/Frequency[i]," ",Harmonic_CL[i]*100," ",Red_Noise_CL[i]*100,"\n")
}
# reassign numpeak
numpeak=numpeak2
# end ar1=T section
}
### generate plots
if(genplot)
{
# first plot power spectrum, with red noise model and confidence levels if requested
if(ar1)
{
par(mfrow=c(3,1))
if(!CLpwr) mtitle=c("MTM Power (black) & AR1 fit (red)")
if(CLpwr) mtitle=c("MTM Power (black); AR1 fit (red); 90%CL, 95%CL, 99%CL (dotted)")
}
if(!ar1)
{
par(mfrow=c(2,1))
mtitle=c("MTM Power")
}
if(pl == 1) logxy="y"
if(pl == 2) logxy=""
if(pl == 3) logxy="xy"
if(pl == 4) logxy="x"
if(pl == 3 || pl == 4) xmin=freq[1]
plot(freq,pwrRaw,type="l", col="black", xlim=c(xmin,xmax), xlab="Frequency",ylab="Power",main=mtitle,cex.axis=1.1,cex.lab=1.1,lwd=2,bty="n",log=logxy)
if(ar1)
{
lines(freq,RawAR,col="red",lwd=2)
if(CLpwr)
{
lines(freq,AR1_90,col="red",lwd=1,lty=3)
lines(freq,AR1_95,col="red",lwd=1,lty=3)
lines(freq,AR1_99,col="red",lwd=1,lty=3)
}
}
### plot "significant" frequencies on power spectrum
if(sigID && (numpeak) > 0)
{
plfreq=double(numpeak)
pltext=double(numpeak)
for (k in 1:(numpeak))
{
plfreq[k]=Frequency[k]
pltext[k]=k
}
abline(v=plfreq,col="gray",lty=3)
mtext(pltext[seq(from=1,to=(numpeak),by=2)], side=3,line=0.25,at=plfreq[seq(from=1,to=(numpeak),by=2)],cex=0.5,font=4)
if((numpeak) > 1) mtext(pltext[seq(from=2,to=(numpeak),by=2)], side=3,line=-0.25,at=plfreq[seq(from=2,to=(numpeak),by=2)],cex=0.5,font=4)
}
# next plot harmonic F-test results or AR1 confidence levels
if(pl == 1) logxy=""
if(pl == 2) logxy=""
if(pl == 3) logxy="x"
if(pl == 4) logxy="x"
if(!ar1)
{
plot(freq,prob*100,type="l",col="red",xlim=c(xmin,xmax),ylim=c(80,100),cex.axis=1.1,cex.lab=1.1,xlab="Frequency",ylab="Confidence Level",main="MTM Harmonic F-Test Confidence Level Estimates",bty="n",lwd=2,log=logxy)
abline(h=c(90,95,99),col="black",lty=3)
if(sigID && numpeak > 0)
{
abline(v=plfreq,col="gray",lty=3)
mtext(pltext[seq(from=1,to=numpeak,by=2)], side=3,line=0.25,at=plfreq[seq(from=1,to=numpeak,by=2)],cex=0.5,font=4)
if(numpeak > 1) mtext(pltext[seq(from=2,to=numpeak,by=2)], side=3,line=-0.25,at=plfreq[seq(from=2,to=numpeak,by=2)],cex=0.5,font=4)
}
}
if(ar1)
{
plot(freq,chiCLRawAR*100,type="l",col="red",xlim=c(xmin,xmax),ylim=c(0,100),cex.axis=1.1,cex.lab=1.1,lwd=2,xlab="Frequency",ylab="Confidence Level",main="AR1 Confidence Level Estimates",bty="n",log=logxy)
abline(h=c(90,95,99),col="black",lty=3)
if(sigID && numpeak > 0)
{
abline(v=plfreq,col="gray",lty=3)
mtext(pltext[seq(from=1,to=numpeak,by=2)], side=3,line=0.25,at=plfreq[seq(from=1,to=numpeak,by=2)],cex=0.5,font=4)
if(numpeak > 1) mtext(pltext[seq(from=2,to=numpeak,by=2)], side=3,line=-0.25,at=plfreq[seq(from=2,to=numpeak,by=2)],cex=0.5,font=4)
}
plot(freq,prob*100,type="l",col="red",xlim=c(xmin,xmax),ylim=c(80,100),cex.axis=1.1,cex.lab=1.1,xlab="Frequency",ylab="Confidence Level",main="Harmonic F-Test Confidence Level Estimates",bty="n",lwd=2,log=logxy)
abline(h=c(90,95,99),col="black",lty=3)
if(sigID && numpeak > 0)
{
abline(v=plfreq,col="gray",lty=3)
mtext(pltext[seq(from=1,to=numpeak,by=2)], side=3,line=0.25,at=plfreq[seq(from=1,to=numpeak,by=2)],cex=0.5,font=4)
if(numpeak > 1) mtext(pltext[seq(from=2,to=numpeak,by=2)], side=3,line=-0.25,at=plfreq[seq(from=2,to=numpeak,by=2)],cex=0.5,font=4)
}
}
# end genplot section
}
if (output==1)
{
if(!ar1)
{
spectrum <- data.frame(cbind(freq,pwrRaw,prob*100))
colnames(spectrum)[1] <- 'Frequency'
colnames(spectrum)[2] <- 'Power'
colnames(spectrum)[3] <- 'Harmonic_CL'
}
if(ar1)
{
spectrum <- data.frame(cbind(freq,pwrRaw,prob*100,chiCLRawAR*100,RawAR,AR1_90,AR1_95,AR1_99))
colnames(spectrum)[1] <- 'Frequency'
colnames(spectrum)[2] <- 'Power'
colnames(spectrum)[3] <- 'Harmonic_CL'
colnames(spectrum)[4] <- 'AR1_CL'
colnames(spectrum)[5] <- 'AR1_fit'
colnames(spectrum)[6] <- 'AR1_90_power'
colnames(spectrum)[7] <- 'AR1_95_power'
colnames(spectrum)[8] <- 'AR1_99_power'
}
return(spectrum)
}
if (output==2)
{
sigfreq <- data.frame(Frequency[1:numpeak])
colnames(sigfreq) <- 'Frequency'
return(sigfreq)
}
if (output==3)
{
sigfreq <- data.frame(Frequency[1:numpeak],Harmonic_CL[1:numpeak])
colnames(sigfreq)[1] <- 'Frequency'
colnames(sigfreq)[2] <- 'Harmonic_CL'
return(sigfreq)
}
if (output==4)
{
return(spec)
}
#### END function mtm
}
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